# Vector line equations

• Jun 16th 2010, 12:35 PM
Stuck Man
Vector line equations
I've done the first part of the attached question. 6lambda:1mu.

I don't think the hints make sense. c is perpendicular to a. How can it be on a plane with a and b and also be parallel to l as implied by r=a+kc?
• Jun 17th 2010, 02:03 PM
slider142
c is not parallel to l. r = a + kc means that each point of l is a resultant of a and some vector collinear with c. This is guaranteed becase a and c form a basis for the plane that l lies in.
This is akin to the vector equation of a line with standard basis: r = b + kx.
• Jun 17th 2010, 02:58 PM
Plato
Quote:

Originally Posted by Stuck Man
I don't think the hints make sense. c is perpendicular to a. How can it be on a plane with a and b and also be parallel to l as implied by r=a+kc?

The vector $a\times(a\times b)=(a\cdot b)a-(a\cdot a)b$ is clearly perpendicular to $a$ and as a linear combination of $a~\&~b$ it is in the plane of $a~,~b,~&~O$.
Now you can finish?
• Jun 17th 2010, 03:10 PM
Plato
Here is what you cannot read
The vector a x (a x b)=(a.b)a-(a.a)b is clearly perpendicular to a and as a linear combination of $a~\&~b$ it is in the plane of $a~,~b,~\&~O$.
• Jun 17th 2010, 03:11 PM
Plato
Here is what you cannot read
The vector a x (a x b)=(a.b)a-(a.a)b is clearly perpendicular to a and as a linear combination of $a~\&~b$ it is in the plane of a, b, & O.
• Jun 18th 2010, 01:26 AM
Stuck Man
Quote:

Originally Posted by slider142
c is not parallel to l. r = a + kc means that each point of l is a resultant of a and some vector collinear with c.

My book says "The equation of a straight line parallel to vector b through a point with position vector a is r=a+tb." That is why I thought the line l would be parallel to c.
• Jun 18th 2010, 05:46 AM
Stuck Man
I have consulted another source which was helpful and can confirm that you are wrong slider142.

I had been thinking that c is a position vector so I can understand it all better now. slider142 also seems to have thought that c is a position vector.

I calculated that lambda:mu is 6:1 but the book says 1:6. Is the book wrong? I think I can do all the rest of the question.
• Jun 19th 2010, 05:09 AM
Stuck Man
After 3 days still no one has given me any help.

Why is lambda:mu 1:6? Is the equation of line l r=i-2j+k+t(13i+4j-5k) as the book says? Why is p=-12i-6j+6k? Why is p not equal to b?
• Jun 20th 2010, 05:08 AM
Stuck Man
I've finally done the question. Line l actually meets OB produced.